Yamaguchi H. Engineering Fluid Mechanics (Fluid Mechanics and Its Applications)
Подождите немного. Документ загружается.
Exercise
The first normal stress coefficient
1
\
is as well computed when
0
2
1
1
2
OK
J
\
N
(5)
Thus,
1
\
is also kept constant for the variation of
J
. The second normal
stress difference
2
N is, however, zero since 0
11
zzyy
CC for the simple
shear flow. Thus, for all
J
, we have
0
2
N
(6)
and
0
2
M
(7)
For the uniaxial stretching, i.e.
0 k in Eq. (7.2.15), with the constant
elongational rate
H
, the elongational viscosity
e
K
is obtained with refer-
ence to Exercise 7.3.1, with the following formulation
>@
HOHO
K
HO
K
O
K
H
O
K
HH
WW
K
O
H
O
H
HHO
O
c
»
¼
º
«
¬
ª
c
c
³
³
³
f
c
c
f
c
c
¸
¹
·
¨
©
§
c
f
¸
¹
·
¨
©
§
c
121
3
1
1
0
11
2
2
0
2
2
0
11
2
02211
t
tttt
t
tttt
tt
t
yyxx
tt
e
tdee
tdeee
tdCCe
(8)
It is mentioned that in Eq. (8)
fo
e
K
as
OH
21o
. Consequently, the re-
sults from (3) to (7) agree with those obtained from the differential UCM
equation, as examined in Eqs. (7.3.85) and (7.3.87).
Exercise 7.3.3 Oldroyd-B Equation
Examine the Oldroyd-B equation for the simple shear flow and shearfree
flow at a steady state.
Ans.
The Oldroyd-B equation given in Eq. (7.3.52) contains two time
constants, namely
1
O
and
2
O
, which are, respectively, the relaxation
485
7 Non-Newtonian Fluid and Flow
time constant and the retardation time constant, with
0
K
being the zero shear
rate viscosity. The rheological predictions of this quasi-linear equation are
useful to expand more admissible constitutive equations. Now let us start
again to consider the steady simple shear flow by writing the equation
»
»
»
¼
º
«
«
«
¬
ª
¸
¸
¸
¹
·
¨
¨
¨
©
§
¸
¸
¸
¹
·
¨
¨
¨
©
§
¸
¸
¸
¹
·
¨
¨
¨
©
§
¸
¸
¸
¹
·
¨
¨
¨
©
§
2
201
000
000
001
2
000
001
010
000
00
02
00
0
0
JOJKW
WW
JO
W
WW
WW
xx
yyxy
zz
yyxy
xyxx
(1)
which gives the following relationships:
JKW
0
xy
,
2
2101
2
JOOKWW
yyxx
N
(2)
and
0
2
zzyy
N
WW
(3)
Resultantly, we can obtain the material functions where
0
KK
,
2101
2
OOK\
and
0
2
\
(4)
As it is obviously seen in comparison with Eq. (3) to (7) in the previ-
ous Exercise of 7.3.2, we have the same material functions as those of the
UCM equation with respect to the steady shear flow.
In the case of a shearfree flow, however, the Oldroyd-B equation can
be written in components form:
»
»
»
¼
º
«
«
«
¬
ª
¸
¸
¸
¹
·
¨
¨
¨
©
§
¸
¸
¸
¹
·
¨
¨
¨
©
§
¸
¸
¸
¸
¸
¸
¸
¹
·
¨
¨
¨
¨
¨
¨
¨
©
§
¸
¸
¸
¹
·
¨
¨
¨
©
§
2
2
2
20
1
100
010
004
100
010
002
1
2
1
00
01
2
1
0
00
2
00
00
00
k
k
k
k
k
k
zz
yy
xx
zz
yy
xx
HOHK
W
W
W
HO
W
W
W
(5)
From such a deduction, each stress component is given where
486
Exercise
HO
HOHK
W
HOHK
HKOHKWHO
1
20
20
2
0201
21
212
212
4221
xx
xx
>@
>@
HO
HOHK
W
HKOHKWHO
1
20
2
2
0201
11
111
1111
k
kk
kkk
yy
yy
and
>@
>@
HO
HOHK
W
HKOHKWHO
1
20
2
2
0201
11
111
1111
k
kk
kkk
zz
zz
Thus, in the case of uniaxial stretching, i.e. 0 k , we can obtain the elon-
gational viscosity from the Oldroyd-B equation, which is
HOHO
HOHOKHOHOK
HO
HOK
HO
HOK
H
WW
K
11
120120
1
20
1
20
121
2111212
1
1
21
212
yyxx
e
HOHO
HOHOKK
11
1200
121
2133
(6)
It is noted that the steady material function in a shearfree flow becomes
infinite at the critical strain rates, i.e.
)2(1
1
H
O
.
Exercise 7.3.4 Nondimensional Parameters in a Viscoelastic Model
Equation
Consider the White-Metzner constitutive equation given in Eq. (7.3.65).
Assuming isothermal flow, nondimensionalize the equation with the fol-
lowing nondimensional parameters
l
x
x
*
,
U
u
u
*
,
0
t
t
t
*
and
n
lU
0
K
IJ
IJ
*
(1)
487
7 Non-Newtonian Fluid and Flow
where l is the characteristic length, U is the characteristic velocity,
0
t is
the reference time, and
0
K
is the zero shear rate viscosity. Set n and
s
to
be power index (constants) for the power law given below
2
1
0
ൖ
2
1
¸
¹
·
¨
©
§
n
e
KOK
(2)
and
2
1
0
ൖ
2
1
¸
¹
·
¨
©
§
ns
e
G
O
JK
O
(3)
Ans.
The constitutive equation is, according to Eq. (7.3.65), written as
e
IJIJ
IJ
IJ
2
1
0
2
1
0
ൖ
2
1
2
ൖ
2
1
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
n
e
T
ns
e
D
D
K
O
uu
t
(4)
Using the relationship in Eq. (1), the constitutive equation is non-
dimensionalized as follows
**
*****
*
*
**
e
IJIJ
IJ
IJ
2
1
2
1
0
2
1
1
0
0
ൖ
2
1
2
ൖ
2
1
ൖ
2
1
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
n
e
T
ns
e
ns
ns
e
ns
l
U
Dt
D
l
U
t
uu
O
O
(5)
We have two nondimensionalized parameters that appear at the second
and the third term of the left hand side of the equation. They are
systemflowofscaletime
timerelaxationfluidofscaletime
1
0
0
¸
¹
·
¨
©
§
flow
ns
tl
U
t
eD
O
O
(6)
488
Problems
and
timesticcharacterisecond
timerelaxation
0
¸
¹
·
¨
©
§
OJO
ns
l
U
We
(7)
where
J
is the characteristic strain rate. eD and We are respectively
called the Deborah number and the Weissenberg number. eD is inter-
preted as dynamic of a polymeric fluid, as the ratio of the magnitude of the
elastic forces to that of the viscous forces, and, in-between eD and We ,
there is the relationship
StWeeD
(8)
where
St is the Strouhal number defined in Eq. (6.2.28) as
0
UtlSt . In
application of the similarity law to dynamic problems of viscoelastic fluid
flow, either eD or
We
can be used to characterize the flow phenomena. In
general,
We
is often used for steady state flow dynamics together with the
Reynolds number
Re . It should be mentioned that the symbol of We is
conventionally used for the Weber number defined in Eq. (6.2.13); thus,
care must be taken to avoid misuse of the symbolism for the Weissenberg
number.
Problems
7.3-1 Sketch the behavior of G
c
, G
cc
,
K
c
and
K
cc
for the Oldroyd-B equa-
tion, when
12
10
OO
. is assumed. What are the physical implications
for the assumption of
12
OO
!
?
Ans.
»
»
»
»
»
¼
º
«
«
«
«
«
¬
ª
stress.
shear theofbehavior
transientheconsider t
flow,shear simpleFor
7.3-2 Explain the physical implications of the Carreau-Yasuda formula,
given in Eq. (7.1.6) from the viewpoint of the CRM equation.
Ans.
»
¼
º
«
¬
ª
n ofindex power
withningShear thin
7.3-3 In treating the same problem found in Eqs. (7.3.113) and (7.3.114),
a method of Laplace transform may be adopted to obtain an ana-
lytical solution. Here the constitutive equation is given by an integral
489
7 Non-Newtonian Fluid and Flow
equation. Derive an ordinary differential equation given by the fol-
lowing form after Laplace transform.
0
4
1
1
,
¸
¹
·
¨
©
§
rus
dr
d
r
dr
d
s
I
]
I
]
where
s
is taken as complex number with mathematical convention,
defining
iy
r
s
, and
x
r
for insuring the convergence of inverse
transformation, otherwise
r
being arbitrary, and
I
is such that
[[I
][
druesr
³
f
f
,,
Ans.
»
»
»
»
»
»
»
»
»
»
¼
º
«
«
«
«
«
«
«
«
«
«
¬
ª
w
w
»
»
»
»
¼
º
«
«
«
«
¬
ª
w
w
°
°
¿
°
°
¾
½
°
°
¯
°
°
®
w
w
c
¸
¸
¹
·
¨
¨
©
§
³
c
transformLaplaceby
B.C.ith equation w this
transform4
1
0
0
,
*
t
u
dt
r
u
er
rr
t
tt
]
O
K
7.3-4 Plot velocity profile of the transient Poiseuille flow of a viscoelastic
fluid (as studied in Section 7.3.3.2 and Problem 7.3-3) and compare
the Newtonian case as indicated in Fig. 7.24. Finite difference cord
developed in Section 7.3, Eqs. (7.3.125) and (7.3.126) are helpful.
Fig. 7.24 Viscoelastic fluid in start-up transient Poiseuille flow
490
Nomenclature
Nomenclature
21
aaa ,,
: constants
21
bbb ,,
: constants
Cc,
: constants
...,,,
321
ccc
: parameters
m
C
: torque coefficient
C
: Cauchy tensor
1
C
: Finger tensor
dD,
: diameter of take rR 22 ,
De
: Deborah number
e
: rate of strain tensor
F
,F
: force vector, thrust force
G
: modulus, relaxation modulus
G
c
: storage modulus
G
cc
: loss modulus
G
: complex modulus
g
G
: Glassy modulus
k
: constant, 10 dd k
c
k
: thermal conductivity
lL
,
: length scale
h
: gap distance
n
: power law index
n
ˆ
: normal unit vector
m
: material constant for power law
21
NN ,
: respectively the first and second normal stress differences
p
: pressure
rR
,
: radius
Q
: flow rate
Re
: Reynolds number
*
Re
: generalized Reynolds number
St
: Strouhal number
s
: power law index, parameter in Laplace transform
t
: time
T
: temperature
r
T
: torque
ȉ
: total stress tensor
U
: bulk (characteristic) velocity, free energy
491
7 Non-Newtonian Fluid and Flow
u
: local velocity
u
: mean (average) flow velocity
W
: scalar (potential) function
c
W
: energy dissipation
We
: Weissenberg number
zy
x
,,
: Cartesian coordinates system
zr ,,
T
: cylindrical coordinates system
M
T
,,r
: spherical coordinates system
D
: mobility factor
E
: gap ratio
J
: strain
J
: shear rate (rate of strain)
a
J
: apparent Newtonian wall shear rate
w
J
: wall shear rate
Ȗ
: strain tensor
R
Ȗ,Ȗ
R
: relative strain tensor
H
: elongational strain
s
H
: stress ratio
H
: elongational (strain) rate
E
H
: tensile (elongational) strain rate
[
]
,
: parameters
K
: viscosity
K
c
: dynamic viscosity
*
K
: complex viscosity
a
K
: apparent viscosity
e
K
: elongational viscosity
0
K
: Newtonian viscosity, zero shear (rate) viscosity
s
K
: solvent contribution viscosity
p
K
: polymer contribution viscosity, plastic viscosity
0
T
T
,
: angle
1
O
O
,
: relaxation time constant,
i
O
: multiple relaxation time conditions
2
O
: retardation time constant
G
: mechanical loss angle (phase-shift)
IJ
: (deviatoric) stress tensor
ij
W
: components of (deviatoric) stress tensor
492
Bibliography
E
W
: net tensile stress
p
IJ
: polymer stress tensor
s
IJ
: solvent stress tensor
w
W
: wall shear stress
I
: mechanical loss angle (phase-shift), scalar function
21
\
\
,
: respectively the first and
second normal stress difference coefficients
Z
: frequency
: rotational speed
Bibliography
The most fundamental treatment of non-Newtonian fluids, in particular to
polymeric liquids, is found in rather authoritative texts, with which every
student of non-Newtonian fluid mechanics should be acquainted:
R.B. Bird, C.F. Curtiss, R.C. Armstrong and O. Hassanger, Dynamics
of Polymeric Liquids, Vol.1 Fluid Mechanics and Vol. 2 Kinetic Theory
(2nd Edition), A Wiley-Interscience Publication, New York, NY, 1987.
R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena
(2nd Edition), John Wiley & Sons, Inc., Hoboken, NJ, 2002.
A lucid mathematical treatment on flows of non-Newtonian fluids, in
which conceptual and logical thinking is developed, is given with
J. Harris, Rheology and Non-Newtonian Flow, Longman Inc., New
York, 1977.
R.R. Huilgol and N. Phan-Thien, Fluid Mechanics of Viscoelasticity,
Elsevier Science Publishers B.V., Amsterdam, 1997.
N.W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic
Behavior, Springer-Verlag, New York, 1989.
Some of flow problems, the boundary layer behavior, are derived from the
reference 3. Some selective topics of flow problems in viscoelastic liquids
with a good deal of literature citation is found in
D.D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag
New York Inc., New York, 1990.
:
2.
3.
4.
5.
6.
7.
1.
R.B. Bird, R.C. Armstrong and O. Hassager, Dynamics of polymeric
liquids, 1. Wiley, New York, 1977.
493
7 Non-Newtonian Fluid and Flow
The current approach in flow problems of non-Newtonian fluids is largely de-
pendent upon numerical analysis with ultrahigh performance computer. A ba-
sic computational viscoelastic fluid dynamic is found in the reference 4, and
R.G. Owens and T.N. Phillips, Computational Rheology, Imperial Col-
lege Press, London, 2002.
M.J. Crochet, A.R. Davies and K. Walters, Numerical Simulation of
Non-Newtonian Flow (3rd Edition), Elsevier Science Publishers B.V.,
Amsterdam, 1991.
Rheological treatment of non-Newtonian fluids and the derivation of the
expression for the constitutive equations are well presented in
F.A. Morrison, Understanding Rheology, Oxford University Press, Inc.,
Oxford, 2001.
11.
R.I. Tanner, Engineering Rheology, Oxford University, Inc., Oxford,
(Reprinted) 1992.
12.
P.J. Carreau, D.C.R. De Kee, and R.P. Chhabra, Rheology of Polymeric
Systems, Hanser/Gardner Publications, Inc., Cincinnati, OH, 1997.
13.
R.G. Larson, Constitutive Equations for Polymer Melts and Solutions,
Butterworths Series in Chemical Engineering, AT&T Bell Laboratories,
Inc., Murray Hill, NJ, 1988
The transport phenomena in simple flows, including the boundary layer
flows in power law fluids, are well presented in the reference 11. A few
texts of standard measurement methods and the contribution of the melt
rheology contain some useful materials on industrial applications. Three
examples are those by
14.
A.A. Collyer and D.W. Clegg, Rheological Measurement, Elsevier
Applied Science Publishers LTD, Amsterdam, 1988.
15.
J.M. Dealy and K.F. Wissbrun, Melt Rheology and Its Role in Plastics
Processing, Kluwer Academic Publishers, Boston, MA (Reprinted) 1999.
16.
J.R.A. Pearson, Mechanics of Polymer Processing, Elsevier Applied
Science Publishers, LTD, Amsterdam, (Reprinted) 1986.
In particular, although not being quoted in the present text, but as one of
current topics in engineering fluid mechanics, the drag reduction of turbu-
lent flows in view of dilute polymeric substance adding is found in
17.
A. Gyr and H.W. Bewersdorff, Drag Reduction of Turbulent Flows by
Additives, Kluwer Academic Publishers, Boston, MA, 1995.
8.
9.
10.
494